P15 - and a force due to air resistence which can be...

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Project 15 MAC 2311 YOU MUST SHOW YOUR WORK TO RECEIVE FULL CREDIT!! 1. Examine the functions f ( x ) = x 2 ln | x | and g ( x ) = e x - 1 x . Evaluate the limits lim x 0 f ( x ) and lim x 0 g ( x ) . Which indeterminate forms do these limits represent? Using the limit deﬁnition of the derivative, show that each of the functions below are differ- entiable (and hence also continuous) at 0 . h ( x ) = x 2 ln | x | x 6 = 0 0 x = 0 k ( x ) = e x - 1 x x 6 = 0 1 x = 0 What do each of the values h 0 (0) and k 0 (0) turn out to be? 2. For each function below, write the indeterminate form that it represents , then evaluate the limit using the techniques learned in class. The answer to each is given to help you. .. a) lim x 0 ± 1 x - 1 e x - 1 ² = 1 2

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b) lim x 0 ( 1 + x + tan(2 x ) ) 1 x = e 3 c) lim x →∞ 1 + ln( x ) ln(1 + x + x 2 ) = 1 2 3. If an object of mass m falls with air resistence, then it experiences two forces: the force of gravity mg
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Unformatted text preview: and a force due to air resistence, which can be accurately modeled as proportional to the velocity v of the object, say-bv for some constant b > . Here, g is the constant acceleration of gravity near the earth’s surface. By Newton’s second law, the force should be equal to the mass of the object times its acceleration dv dt . a) Show that the function v ( t ) = mg b ± 1-e-bt/m ² satisﬁes m dv dt = mg-bv for all t . b) Calculate lim t →∞ v ( t ) . What does it imply about the velocity of the object as time passes? c) For any ﬁxed (constant) time t , calculate lim b → + v ( b ) . What does it imply about an object falling without air resistence?...
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P15 - and a force due to air resistence which can be...

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